Circles
Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segments joining the points of contact at the centre.
Prove that the opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC.
Or
A quadrilateral is drawn to circumscribe a circle. Prove that the sum of opposite sides are equal.
Two concentric circles are of radii 5 cm and 3 cm respectively. Find the length of the chord of the larger circle which touches the smaller circle.
Prove that the parallelogram circumscribing a circle, is a rhombus.
Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.
If two tangents are drawn to a circle from an external point, show that they subtend equal angles at the centre.
In the given figure, if AB = AC, prove that BE = CE.
Prove that the tangents drawn at the ends of the diameter of a circle are parallel.
Prove that the lengths of two tangents drawn from an external point to a circle are equal.
Fill in the blanks.
(i) A line intersecting a circle in two distinct points is called a ………
(ii) A circle can have parallel tangents at the most …
(iii) The common point of a tangent to a circle and the circle is called the ………
(iv) A circle can have ….. tangents
Sol: A line intersecting a circle at two district points is called a secant A circle can have two parallel tangents at the most The common point of a tangent to a circle and the circle is called the...
Two tangents segments BC and BD are drawn to a circle with center O such that CBD 120. Prove that OB 2BC
In the given figure, PA and PB are two tangents form an externa point P to a circle with centre O. If PBA 65 , find the OAB and APB.
In the given figure, PA and PB are the tangents to a circle with centre O. Show that the points A, O, B, P are concyclic.
In the given figure, a quadrilateral ABCD is drawn to circumscribe a circle such that its sides AB, BC, CD and AD touch the circle at P, Q, R and S respectively. If AB = x cm, BC = 7cm, CR = 3cm and AS=5cm, find x.
In the given figure, AD and AE are the tangents to a circle with centre O and BC touches the circle at F. If AE = 5 cm, then perimeter of ABC is
If tangents PA and PB from a point P to a circle with center O are drawn so that APB 80, then, POA?
If the angles between two radii of a circle is 130 , then the angle between the tangents at the ends of the radii is (a) 65 (b) 40 (c) 50 (d) 90
In the given figure, O is the center of a circle, PQ is a chord and the tangent PT at P makes an angle of 50 with PQ. Then, POQ ?
Which of the following statement is not true?
(a) A line which intersect a circle in tow points, is called secant of the circle.
(b)A line intersecting a circle at one point only, is called a tangent to the circle.
(c) The point at which a line touches the circle, is called the point of contact.
(d) A tangent to the circle can be drawn form a point inside the circle.
Answer: (d) A tangent to the circle can be drawn form a point inside the circle. Sol: A tangent to the circle can be drawn from a point Inside the circle. This statement is false because tangents...
Which of the following statements is not true?
(a) A tangent to a circle intersects the circle exactly at one point.
(b) The point common to the circle and its tangent is called the point of contact.
(c) The tangent at any point of a circle is perpendicular to the radius of the circle through the point of contact.
(d) A straight line can meet a circle at one point only.
Answer: (d) A straight line can meet a circle at one point only. Sol: A straight be can meet a circle at one point only This statement is not true because a straight line that is not a tangent but a...
Which of the following statements in not true?
(a)If a point P lies inside a circle, not tangent can be drawn to the circle, passing through p.
(b) If a point P lies on the circle, then one and only one tangent can be drawn to the circle at P.
(c) If a point P lies outside the circle, then only two tangents can be drawn to the circle form P.
(d) A circle can have more than two parallel tangents. parallel to a given line.
Answer: (d) A circle can have more than two parallel tangents. parallel to a given line. Sol: A circle can have more than two parallel tangents. parallel to a given line. This statement is false...
In the given figure, O is the centre of two concentric circles of radii 5 cm and 3 cm. From an external point p tangents PA and PB are drawn to these circles. If PA = 12 cm then PB is equal to
In the given figure, AP, AQ and BC are tangents to the circle. If AB = 5 cm, AC = 6 cm and BC = 4 cm then the length of AP is
In the given figure, three circles with centres A, B, C respectively touch each other externally. If AB = 5 cm, BC = 7 cm and CA = 6 cm then the radius of the circle with centre A is
In the given figure, DE and DF are tangents from an external point D to a circle with centre A. If DE = 5 cm and DE DF then the radius of the circle is (a) 3 cm (b) 4 cm (c) 5 cm (d) 6 cm
In the given figure, PA and PB are tangents to the given circle such that PA = 5 cm and APB 60 . The length of chord AB is
Quadrilateral ABCD is circumscribed to a circle. If AB= 6 cm, BC = 7cm and CD = 4cm then the length of AD is (a) 3 cm (b) 4 cm (c) 6 cm (d) 7 cm
In the given figure, ABC is right-angled at B such that BC = 6 cm and AB = 8 cm. A circle with centre O has been inscribed the triangle. OP AB,OQ BCand OR AC. If OP = OQ = OR = x cm then x = ?
In the given figure, a circle is inscribed in a quadrilateral ABCD touching its sides AB, BC, CD and AD at P, Q, R and S respectively. If the radius of the circle is 10 cm, BC = 38 cm, PB = 27 cm and AD CD then the length of CD is
In a right triangle ABC, right angled at B, BC = 12 cm and AB = 5 cm. The radius of the circle inscribed in the triangle is (a) 1 cm (b) 2 cm (c) 3 cm (d) 4 cm
In the given figure, O is the centre of a circle, AB is a chord and AT is the tangent at A. If AOB 100 then BAT is equal to
In the given figure, quad. ABCD is circumscribed, touching the circle at P, Q, R and S. If AP = 6 cm, BP = 5 cm, CQ = 3 cm and DR = 4 cm then perimeter of quad. ABCD is
In the given figure, quad. ABCD is circumscribed touching the circle at P, Q, R and S. If AP = 5 cm, BC= 7 c m and CS = 3 cm. Then, the length of AB = ?
In the given figure, QR is a common tangent to the given circles, touching externally at the point T. The tangent at T meets QR at P. If PT= 3.8 cm then the length of QR is
In the given figure, a triangle PQR is drawn to circumscribe a circle of radius 6 cm such that the segments QT and TR into which QR is divided by the point of contact T, are of lengths 12 cm and 9 cm respectively. If the area of PQR 189 cm2 then the length of side of PQ is (a) 17.5 cm (b) 20 cm (c) 22.5 cm (d) 25 cm
In the given figure, O is the centre of a circle; PQL and PRM are the tangents at the points Q and R respectively and S is a point on the circle such that SQL 50 and DE DF OQ BC and OR AC.
To draw a pair of tangents to a circle, which are inclined to each other at an angle of 45, we have to draw tangents at the end points of those two radii, the angle between which is (a) 105 (b) 135 (c) 140 (d) 145
In the given figure, a circle touches the side DF of EDF at H and touches ED and EF produced at K and M respectively. If EK = 9 cm then the perimeter of EDF is
In the given figure, O is the centre of a circle, BOA is its diameter and the tangent at the point P meets BA extended at T. If PBO 30 then PTA ?
The length of the tangent form an external point P to a circle of radius 5 cm is 10 cm. The distance of the point from the centre of the circle is (a) 8 cm (b)
In the given figure, PQR is a tangent to the circle at Q, whose centre is O and AB is a chord parallel to PR such that BQR 70 . Then, AQB ? (a) 20 (b) 35 (c) 40 (d) 45
O is the centre of a circle of radius 5 cm. At a distance of 13 cm form O, a point P is taken. From this point, two tangents PQ and PR are drawn to the circle. Then, the area of quad. PQOR is
In the given figure, two circles touch each other at C and AB is a tangent to both the circles. The measure of ACB is
In the given figure, PQ is a tangent to a circle with centre O, A is the point of contact. If PAB 67 , then the measure of
In the given figure, O is the centre of the circle. AB is the tangent to the circle at the point P. If PAO 30 then CPB ACP is equal to
In the given figure, O is the centre of the circle. AB is the tangent to the circle at the point P. If APQ 58 then the measure of PQB is
If PA and PB are two tangents to a circle with centre O such that APB 80 . Then, AOP ?
In the given figure, PQ and PR are tangents to a circle with centre A. If QPA 27 then QAR equals
In the given figure, PQ and PR are tangents to a circle with centre A. If QPA 27 then QAR equals
If two tangents inclined at an angle of 60 are drawn to a circle of a radius 3 cm then the length of each tangent is
In the given figure, PA and PB are two tangents to th4e circle with centre O. If APB 60 then OAB is
In the given figure, O is the centre of a circle and PT is the tangent to the circle. If PQ is a chord such that QPT 50then POQ ?
In the given figure, If AOD 135 then BOC equal to
In the given figure, the length of BC is
If PA and PB are two tangents to a circle with centre O such that AOB 110 then APB is equal to
In the given figure, AT is a tangent to the circle with center O such that OT = 4 cm and OTA 30, Then, AT ?
In the given figure, O is the center of a circle, PQ is a chord and Pt is the tangent at P. If POQ 70 , then TPQ is equal to
In the given figure, 0 is the centre of a circle, AOC is its diameter such that ACB 50. If AT is the tangent to the circle at the point A, then BAT ?
In the given figure, AB and AC are tangents to a circle with centre O and radius 8 cm. If OA=17 cm, then the length of AC (in cm) is
In the given figure, O is the centre of two concentric circles of radii 6 cm and 10 cm. AB is a chord of outer circle which touches the inner circle. The length of chord AB is
If a chord AB subtends an angle of 60 at the center of a circle, then he angle between the tangents to the circle drawn form A and B is (a) 30 (b) 60 (c) 90 (d) 120
In the given figure, AB and AC are tangents to the circle with center O such that BAC 40 . Then , BOC = 40.
PQ is a tangent to a circle with centre O at the point P. If OQP is equal to
In the given figure, point P is 26 cm away from the center O of a circle and the length PT of the tangent drawn from P to the circle is 24 cm. Then, the radius of the circle is
In the given figure, PT is tangent to the circle with centre O. If OT = 6 cm and OP = 10 cm then the length of tangent PT is
The chord of a circle of radius 10 cm subtends a right angle at its centre. The length of the chord (in cm) is
Which of the following pairs of lines in a circle cannot be parallel?
(a) two chords (b) a chord and tangent (c) two tangents (d) two diameters
Answer: (d) two diameters Sol: Two diameters cannot be parallel as they perpendicularly bisect each other.
In a circle of radius 7 cm, tangent PT is drawn from a point P such that PT =24 cm. If O is the centre of the circle, then length OP = ?
(a) 30 cm (b) 28 cm (c) 25 cm (d) 18 cm Answer: (c) 25 cm Sol: The tangent at any point of a circle is perpendicular to the radius at the...
In the given figure, RQ is a tangent to the circle with centre O, If SQ = 6 cm and QR = 4 cm. then OR is equal to
The number of tangents that can be drawn form an external point to a circle is (a) 1 (b) 2 (3) (d) 4
In the given figure, PA and PB are two tangents to the circle with centre O. If APB 60 , then find the measure of OAB.
13. In the given figure, PQ is chord of a circle with centre O an PT is a tangent. If QPT 60, find the PRQ.
In two concentric circles, a chord of length 8cm of the large circle touches he smaller circle. If the radius of the larger circle is 5cm then find the radius of the smaller circle.
In the given figure, O is the centre of the circle. PA and PB are tangents. Show that AOBP is cyclic quadrilateral.
In the given figure, a cradle inscribed in a triangle ABC touches the sides AB, BC and CA at points D, E and F respectively. If AB = 14cm, BC = 8cm and CA=12 cm. Find the length AD, BE and CF.
BD = 5cm = BE Solving (3) and (4), we get and AD = 9cm
In the given figure, two tangents RQ, and RP and RP are drawn from an external point R to the circle with centre O. If PRQ 120 , then prove that OR = PR + RQ.
Prove that the perpendicular at the point of contact of the tangent to a circle passes through the centre. Sol:
Two concentric circles are of radii 5cm and 3cm. Find the length of the chord of the larger circle (in cm) which touches the smaller circle.
In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 2 cm such that the segments BD and DC into which BC is divided by the point of contact D, are of lengths 4cm and 3cm respectively. If the area of sides AB and AC.
If PT is a tangent to a circle with center O and PQ is a chord of the circle such that QPT 70, then find the measure of POQ.
In the given figure common tangents AB and CD to the two circles with centres O1 and O2 intersect at E. Prove that AB=CD.
In the given figure, O is the centre of a circle. PT and PQ are tangents to the circle from an external point P. If TPQ 70 , find the TRQ.
In the given figure, PA and PB are two tangents to the circle with centre O. If APB 50 then what is the measure of OAB.
In the adjoining figure, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB=6cm, BC=9cm and CD=8 cm. Find the length of side AD.
Sol: We know that when a quadrilateral circumscribes a circle then sum of opposites sides is equal to the sum of other opposite sides. \ AB + CD = AD + BC Þ 6 + 8 = AD = 9 Þ AD = 5 cm...
In the given figure, O is the centre of the circle and TP is the tangent to the circle from an external point T. If PBT 30 , prove that BA : AT = 2 : 1.
In the given figure, a circle with center O, is inscribed in a quadrilateral ABCD such that it touches the side BC, AB, AD and CD at points P, Q, R and S respectively. If AB = 29cm, AD = 23cm, B 90 and DS=5cm then find the radius of the circle.
Prove that the line segment joining the points of contact of two parallel tangents of a circle, passes through its centre.
PQ is a chord of length 4.8 cm of a circle of radius 3cm. The tangents at P and Q intersect at a point T as shown in the figure. Find the length of TP.
In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 3 cm such that the segments BC and DC into which BC is divided by the point of contact D, are of lengths 6cm and 9cm respectively. If the area of sides AB and AC. ABC 54cm2 then find the lengths of
In the given figure, O is the centre of the two concentric circles of radii 4 cm and 6cm respectively. AP and PB are tangents to the outer and inner circle respectively. If PA = 10cm, find the length of PB up to one place of the decimal.
In the given figure, an isosceles triangle ABC, with AB = AC, circumscribes a circle. Prove that point of contact P bisects the base BC.
In the given figure, PA and PB are the tangent segemtns to a circle with centre O. Show that he points A, O, B and P are concyclic.
A circle is inscribed in a ABC touching AB, BC and AC at P, Q and R respectively. If AB = 10 cm, AR=7cm and CR=5cm, find the length of BC.
From an external point P, tangents PA and PB are drawn to a circle with center O. If CD is the tangent to the circle at a point E and PA = 14cm, find the perimeter of PCD
In the given figure, the chord AB of the larger of the two concentric circles, with center O, touches the smaller circle at C. Prove that AC = CB.
In the given figure, a circle touches all the four sides of a quadrilateral ABCD whose three sides are AB = 6cm, BC=7cm and CD=4 cm. Find AD.
In the given figure, a circle inscribed in a triangle ABC, touches the sides AB, BC and AC at points D, E and F Respectively. If AB= 12cm, BC=8cm and AC = 10cm, find the length of AD, BE and CF.
Two concentric circles are of radii 6.5 cm and 2.5 cm. Find the length of the chord of the larger circle which touches the smaller circle.
A point P is 25 cm away from the center of a circle and the length of tangent drawn from P to the circle is 24 cm. Find the radius of the circle.
Find the length of tangent drawn to a circle with radius 8 cm form a point 17 cm away from the center of the circle
Draw two concentric circles of radii 4 cm and 6 cm. Construct a tangent to the smaller circle from a point on the larger circle. Measure the length of this tangent.
In Fig below, PQ is tangent at point R of the circle with center O. If ∠TRQ = 30°, find ∠PRS
Given, $\angle TRQ={{30}^{\circ }}$ . At point R, OR ⊥ RQ. So, $\angle ORQ={{90}^{\circ }}$ $\Rightarrow \angle TRQ+\angle ORT={{90}^{\circ }}$ $\Rightarrow \angle ORT={{90}^{\circ...
If AB, AC, PQ are the tangents in the figure, and AB = 5 cm, find the perimeter of ∆APQ
Since AB and AC are the tangents from the same point A ∴AB=AC=5cm Similarly, BP=PX and XQ=QC Perimeter of \[\Delta APQ=AP+AQ+PQ\] \[=AP+AQ+(PX+XQ)\] \[=(AP+PX)+(AQ+XQ)\] \[=(AP+BP)+(AQ+QC)\]...
A chord PQ of a circle is parallel to the tangent drawn at a point R of the circle. Prove that R bisects the arc PRQ.
Provided in question: Chord PQ is parallel to tangent at R.To prove: R bisects the arc PRQ. Proof: Since PQ || tangent at R. $\angle 1=\angle 2$ [alternate interior angles]$\angle 1=\angle 3$...
Out of the two concentric circles, the radius of the outer circle is 5 cm and the chord AC of length 8 cm is a tangent to the inner circle. Find the radius of the inner circle.
Suppose C1 and C2 are two circles with the same center O. And AC is a chord touching C1 at the point D let’s join OD.So, $OD\bot AC$$AD=DC=4cm$ [perpendicular line OD...
If the quadrilateral sides touch the circle, prove that sum of pair of opposite sides is equal to the sum of other pair.
Let’s Consider a quadrilateral ABCD touching circle with the centre O at points E, F, G and H as we can see in figure. We know that, In a circle with two points outside of it, the tangents drawn...
If PT is a tangent at T to a circle whose centre is O and OP = 17 cm, OT = 8 cm. Find the length of the tangent segment PT.
Given in the question, OT = radius = $8cm$ OP = $17cm$ It is given to find: PT = length of tangent =$?$ T is point of contact. We also know that the tangent and radius are perpendicular at the point...
Fill in the blanks:
(i) A circle can have …………… parallel tangents at the most. (ii) The common point of a tangent to a circle and the circle is called ………… Answer: (i) A circle can have two parallel tangents...
Fill in the blanks:
(i) A tangent to a circle intersects it in …………… point(s). (ii) A line intersecting a circle in two points is called a …………. Answer: (i) A tangent to a circle intersects it...
Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
Answer: First, draw a quadrilateral ABCD that circumscribes a circle with centre O, touching the circle at points P, Q, R, and S. We now have the following figure after joining...
A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively (see Fig. 10.14). Find the sides AB and AC.
Answer: The given figure: Considering the △ABC, We know that any two tangents drawn from the same point to the circle have the same length. As a result, (i) BE = BD = 8 cm (ii) CF = CD = 6 cm (iii)...
Prove that the parallelogram circumscribing a circle is a rhombus.
Answer: Consider the parallelogram ABCD, which circumscribes a circle with O as centre. As ABCD is a parallelogram, so AB = CD and BC = AD. As seen in the figure above, (i) BP = BQ (ii) DR...
Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the center.
Answer: First, draw a circle with the centre O. Draw two tangents PA and PB at point A and point B, respectively, from an exterior point P. Now join A and B to form AB in such a way that it subtends...
In Fig. 10.13, XY and X′Y′ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X′Y′ at B. Prove that ∠ AOB = 90°.
Answer: From the given figure in the textbook, join OC. The diagram will now be as- Now using the SSS congruency the triangles △OPA and △OCA are similar: (i) OP = OC since they are the same circle’s...
A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove that AB + CD = AD + BC
Answer: The given figure is: We can draw a few conclusions from this figure, which are as follows: (i) BP = BQ (ii) DR = DS (iii) CR = CQ (iv) AP = AS Since the above drawn conclusions are tangents...
Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
Answer: With the centre O, draw two concentric circles. Now, in the larger circle, draw a chord AB that touches the smaller circle at a point P, as shown in the diagram below. AB is tangent to the...
The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.
Answer: Draw a diagram according to the question. Here, AB is a tangent drawn on the circle from a point A. As a result, OB will be perpendicular to AB i.e. OB ⊥ AB We know that, AB =...
Prove that the perpendicular at the point of contact to the tangent to a circle passes through the center.
Solution: Draw a circle with a centre O and a tangent AB that touches the circle's radius at point P. To Prove: PQ passes through point O. Consider the case where PQ does not pass through point...
Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
Answer: Firstly, draw a circle and connect two points A and B such that AB becomes the circle's diameter. Now, at points A and B, draw two tangents PQ and RS, respectively. Both radii, AO and...
If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then ∠ POA is equal to
(A) 50° (B) 60° (C) 70° (D) 80° Answer: First, construct a diagram according to the statement given. Now, in the diagram above, OA represents the radius to tangent PA, while OB represents the radius...
In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that ∠POQ = 110°, then ∠PTQ is equal to
(A) 60° (B) 70° (C) 80° (D) 90° Answer: The radius of the circle to the tangent PT is OP, and the radius to the tangents TQ is OQ, as stated in the question. As a result, OP ⊥ PT and TQ ⊥ OQ...
From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is
(A) 7 cm (B) 12 cm (C) 15 cm (D) 24.5 cm Answer: First draw a perpendicular from the triangle's centre O to a point P on the circle that touches the tangent. This line will be perpendicular to the...
Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.
Answer: XY and AB are two parallel lines in the figure above. The line segment AB is the tangent at point C, while the secant is line segment XY.
A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is :
(A) 12 cm (B) 13 cm (C) 8.5 cm (D) √119 cm Answer: In the figure above, the line drawn from the given circle's centre to the tangent PQ is perpendicular to PQ. As a result, OP ⊥ PQ In triangle ΔOPQ,...
How many tangents can a circle have?
Answer: A circle can have an infinite number of tangents. A circle is made up of an infinite number of points that are all at the equal distance from a single point. Infinite tangents can be...